Poisson Fibrations and Fibered Symplectic Groupoids

Contemporary Mathematics

Olivier Brahic and Rui Loja Fernandes

Abstract. We show that Poisson fibrations integrate to a special kind of symplectic fibrations, called fibered symplectic groupoids.

1. Introduction Our purpose in this paper is to explain that there is a geometric theory of Pois- son fibrations that is analogous to the theory of symplectic fibrations. This paper makes no special claims to originality, and improves on previous works of Vorobjev (Poisson case, [15]) and Vaisman (Dirac case, [14]). Our main contribution is two folded. On one hand, we propose an approach based on gauge theory and Dirac geometry, which gives some natural explanations for some of the mysterious formu- las that appear in those works. On the other hand, we look for the first time into the integration of such structures, recovering symplectic fibrations from Poisson fibrations. A symplectic fibration is a locally trivial fibration with fiber type a symplectic manifold, admitting a collection of trivializations whose transition functions are symplectomorphisms. Symplectic fibrations have a long history going back to the early works of Weinstein et al. [18, 10] and Guillemin et al. ([12, 11]). A closed 2-form on the total space of a fibration which restricts to a symplectic form on the fibers, determines a symplectic fibration. Conversely, given a symplectic fibration, one may ask if there exists a closed 2-form which is compatible with the fibration. It is well-known that there are non-trivial obstructions for the existence of such coupling forms and that one can classify all such coupling forms. A very nice expo- sition of the the theory of symplectic fibrations, where these results are discussed in detail, is given in Chapter 6 of the monograph by McDuff and Salamon [13]. We are interested in the more general notion of a Poisson fibration, i.e., a locally trivial fibration with fiber type a Poisson manifold, admitting a collection of trivializations whose transition functions are Poisson diffeomorphisms. At first sight, the analogous questions for Poisson geometry are either hopeless or trivial. On one hand, there are simple examples of fibrations with a Poisson structure on

1991 Mathematics Subject Classiﬁcation. Primary 53D17; Secondary 58H05. Key words and phrases. Poisson and symplectic ﬁbrations; symplectic groupoid. Supported in part by FCT/POCTI/FEDER and by grants POCI/MAT/57888/2004 and POCI/MAT/55958/2004.

c 0000 (copyright holder) 1 2 OLIVIER BRAHIC AND RUI LOJA FERNANDES the total space that restricts to each fiber and which is not a Poisson fibration. On the other hand, every Poisson fibration always admits trivially a compatible Poisson structure: one can just declare the fibers to be Poisson submanifolds. A closer inspection, however, reveals a very different point of view. Note that for a symplectic fibration one looks for a presymplectic structure which intersects each fiber in a symplectic submanifold. Therefore, for a Poisson fibration we should look for a Dirac structure for which each presymplectic leaf intersects every fiber in a symplectic leaf of the Poisson structure on the fiber (recall that a Poisson manifold is a (singular) foliation by symplectic manifolds, while a Dirac manifold is a (singular) foliation by presymplectic manifolds). This yields the notion of a Dirac coupling for a Poisson fibration. The theory of Poisson fibrations can then be seen as a foliated version of the theory of symplectic fibrations (this is the point of view advocated by Vaisman [14]). Therefore, the questions (and answers) in the theory of Poisson fibrations, are analogous to (and generalize) the theory of symplectic fibrations. For example, given a Poisson fibration one would like to know (i) if it admits coupling Dirac structures and (ii) classify all such couplings. In this direction we have the following generalization of a well-known result in symplectic fibrations (see [13, Theorem 6.13]): Theorem 1.1. Let p : M → B be a Poisson fibration. Then the following statements are equivalent: (i) p : M → B admits a coupling a Dirac structure. (ii) There exists a Poisson connection on p : M → B whose holonomy groups act on the fibers in a hamiltonian fashion. Its is also possible to classify all such coupling forms. The proof of this result follows the same pattern as in the symplectic case: one builds a Poisson gauge theory where coupling forms are obtained on associated fiber bundles M = P ×G F , starting from a connection on a principal G-bundle and a Hamiltonian G-action on a Poisson manifold (F, π). Note that G is not necessarily a finite dimensional Lie group. Our second main purpose is the integration of Poisson fibrations. Recall that Poisson structures are infinitesimal objects which integrate to global objects called symplectic groupoids. There are obstructions to integrability which were recently understood [4, 3], but we will ignore them for the time being. Now, we will see that: • The global object associated to a Poisson fibration is a fibered symplectic groupoid. Let us explain what we mean by this. By a fibered groupoid we mean a groupoid G ⇒ M which is fibered over B: G / M ? / ?? }} ? }} ?? }} ? ~}} B Here, G and B are fibered as well as all structure maps. So a fibered groupoid maybe thought of as a fiber bundle with fiber type a groupoid where the structure group acts by groupoid automorphism. Then by a fibered symplectic groupoid we POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 3 mean a fibered groupoid G whose fiber type is a symplectic groupoid. For a fibered symplectic groupoid, G → B is a symplectic fibration and the symplectic structure is compatible with the groupoid structure. The base p : M → B of a fibered symplectic groupoid has a natural structure of a Poisson fibration. Moreover, the fibers of G → B are symplectic groupoids over the fibers of p : M → B, inducing the Poisson structures on the fibers. In particular, the fibers p : M → B are integrable Poisson manifolds. Conversely, we will show the following: Theorem 1.2. Let p : M → B be a Poisson fibration with fiber type (F, π) an integrable Poisson manifold. There exists a unique (up to isomorphism) source 1-connected fibered symplectic groupoid integrating p : M → B. Moreover, this symplectic fibration always admits a coupling 2-form. It was proved in [2] that the global objects integrating Dirac structures are presymplectic groupoids. Therefore, if p : M → B is a Poisson fibration which admits a coupling Dirac structure, there are two natural groupoids associated with it: (i) The source 1-connected fibered symplectic groupoid integrating p : M → B. (ii) The source 1-connected presymplectic groupoid integrating the coupling Dirac structure of p : M → B. The precise relationship between these two objects is more involved, and it would take us too far afield, so it will be discussed elsewhere.

2. Connections and Dirac structures One can define a connection on a fibration by specifying on the total space an almost Dirac structure of a special type. In this section we make a preliminary study of connections on fibrations induced by Dirac structures. 2.1. Connections defined by almost Dirac structures. In what follows, by a fibration we always mean a locally trivial fiber bundle. Given such a fibration p : M → B, we denote by Fb the fiber over b ∈ B, and we let Vert ⊂ TM be the vertical sub-bundle whose fibers are Vert x ≡ Ker dxp = TxFp(x). By a connection Γ on p : M → B we mean an Ehresmann connection, i.e., a distribution Γ : x 7→ Hor x in TM which splits the tangent bundle as

(2.1) TxM = Hor x ⊕ Vert x, and satisﬁes the following lifting property:

Lifting Property: for each x0 ∈ M and each curve γ : [0, 1] → B starting at b0 = p(x0) there exists an integral curve γe : [0, 1] → M of Γ starting at x0 and covering γ. 1 Remark 2.1. Given a C -path γ : [0, 1] → B and x ∈ Fγ(0) the splitting (2.1) guarantees that there exists a unique horizontal lift γex : [0, ε] → M starting at x. The Lifting Property says that we can take ε = 1 (1). In the usual way, one obtains the notion of parallel transport of ﬁbers: given a piecewise C1-path on the base γ : [0, 1] → B we have the diﬀeomorphism:

φγ (t): Fγ(0) → Fγ(t), x 7→ γex(t).

1Note that our curves are always parameterized in the interval [0, 1]. 4 OLIVIER BRAHIC AND RUI LOJA FERNANDES

When γ is a loop, we call φγ (1) the holonomy of Γ along γ. For b ∈ B, the holonomy group with base point b is the subgroup Φ(b) ⊂ Diff (Fb) formed by all holonomy transformations φγ (1), where γ : [0, 1] → B is any loop based at b. Note that to define the concatenation of loops we need to reparameterize our curves, but this causes no problem since “horizontal lift commutes with concatenation” and, hence, two paths differing by a reparameterization determine the same holonomy transformation. Now our basic observation is that connections can be defined by specifying on the total space of the fibration an almost Dirac structure of a special type. Namely:

Definition 2.2. Let p : M → B be a fibration. An almost Dirac structure L on the total space of the fibration is called fiber non-degenerate if

(2.2) (Vert ⊕ Vert 0) ∩ L = {0}.

In fact, we have:

Proposition 2.3. Let p : M → B be a fibration and L a fiber non-degenerate almost Dirac structure. Then L defines a connection ΓL with horizontal space: (2.3) Hor = X ∈ TM : ∃α ∈ (Vert )0, (X, α) ∈ L .

Moreover, any connection on p : M → B can be obtained in this way.

Proof. Take a vector space W , with a maximal isotropic subspace L ⊂ W × W ∗, and let V ⊂ W be a subspace such that:

(V × V 0) ∩ L = {0}.

We claim that W = V ⊕ H, 0 where H = w ∈ W : ∃ξ ∈ V , (w, ξ) ∈ L . If we take W = TxM, L = Lx and V = Vert x, this claim yields the ﬁrst part of the proposition. To prove the claim, we start by checking that V ∩ H = {0}. In fact, if v ∈ H there exists ξ ∈ V 0 such that (v, ξ) ∈ L. Hence, if v ∈ V ∩ H we obtain:

(v, ξ) ∈ (V × V 0) ∩ L = {0}, so we must have v = 0. It remains to check that W = V + H. First, we observe that since L and V × V 0 intersect trivially and dim L = dim(V × V 0) = dim W , we must have: W × W ∗ = (V × V 0) ⊕ L. Therefore, for any w ∈ W we have a decomposition:

(w, 0) = (v, ξ) + (h, η), where v ∈ V , ξ ∈ V 0 and (h, η) ∈ L. It follows that η = −ξ ∈ V 0, so that h ∈ H. We conclude that w = v + h, with v ∈ V and h ∈ H, as claimed. Conversely, if Γ is a connection with horizontal distribution Hor , then L = Hor ⊕Hor 0 deﬁnes a ﬁber non-degenerate almost Dirac structure whose associated connection is Γ. Hence, every connection arises in this way. POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 5

2.2. The horizontal 2-form and the vertical bivector field. Note that two fiber non-degenerate almost Dirac structures L1 and L2 may lead to the same connection ΓL1 = ΓL2 . In fact, as we will see now, there is more structure associated with the specification of a fiber non-degenerate almost Dirac structure. Proposition 2.4. Let p : M → B be a fibration and L a fiber non-degenerate almost Dirac structure, with associated connection ΓL. Then the horizontal distribution Hor is contained in the characteristic distribution of L, and the pull-back of 2 the natural 2-form yields a smooth 2-form ωL ∈ Ω (Hor ).

We will refer to ωL as the horizontal 2-form of L. Proof. Fix a fiber non-degenerate almost Dirac structure L on the total space of a fibration p : M → B. Relations (2.2) and (2.3) together show that, for each X ∈ Hor , there exists a unique α ∈ Vert 0 such that (X, α) ∈ L. One can then define a skew-symmetric bilinear form ω : Hor × Hor → R by: 1 (2.4) ω (X ,X ) := (α (X ) − α (X )) , L 1 2 2 1 2 2 1 0 with α1, α2 ∈ Vert the unique elements such that (X1, α1), (X2, α2) ∈ L. Since L is maximal isotropic we have:

0 = 2h(X1, α1), (X2, α2)i+ = α1(X2) − α2(X1), so this two form can also be written:

(2.5) ωL(X1,X2) = α1(X2) = −α2(X1). 2 In this way we obtain a smooth 2-form ωL ∈ Ω (Hor ). From the definition (2.3) of Hor it is clear that the horizontal distribution Hor is contained in the characteristic distribution of L. From (2.5), it is clear that ωL is the pull-back of the natural 2-form on the characteristic distribution of L. This construction of the horizontal 2-form can be dualized: Proposition 2.5. Let p : M → B be a fibration and let L be a fiber non- degenerate almost Dirac structure, with associated connection ΓL. For each fiber −1 ∗ i : Fx = p (x) ,→ M the pull-back almost Dirac structure i L is well-defined and 2 coincides with the graph of a bivector field πL ∈ X (Vert ).

We will refer to πL as the vertical bivector ﬁeld of L. Proof. First observe that the annihilator of the horizontal space is: (2.6) Hor 0 = {α ∈ T ∗M : ∃X ∈ Vert , (X, α) ∈ L} . Relations (2.2) and (2.6) together show that, for each α ∈ Hor 0, there exists a unique X ∈ Vert such that (X, α) ∈ L. One can then deﬁne a skew-symmetric 0 0 bilinear form πL : Hor × Hor → R by: 1 (2.7) π (α , α ) := (α (X ) − α (X )) , L 1 2 2 1 2 2 1 with X1,X2 ∈ Vert the unique elements such that (X1, α1), (X2, α2) ∈ L. Since L is maximal isotropic we have:

0 = 2h(X1, α1), (X2, α2)i = α1(X2) − α2(X1), 6 OLIVIER BRAHIC AND RUI LOJA FERNANDES

0 0 the form πL : Hor × Hor → R can also be written:

(2.8) πL(α1, α2) = α1(X2) = −α2(X1). Now we remark that the splitting TM = Hor ⊕ Vert allows us to identify Hor 0 = ∗ Vert , so πL becomes a bivector field on the fibers of p : M → B. −1 Let us fix a fiber i : Fx = p (x) ,→ M. Then TFx = Vert and we identify ∗ ∗ 0 ∗ T Fx = Vert ' Hor . The pull-back Dirac structure i L is then given by: ∗ ∗ i L = {(X, α|Vert ) ∈ Vert ⊕ Vert :(X, α) ∈ L} 0 = (X, α) ∈ Vert ⊕ Hor : X = πL(α, ·) = graph(πL), where for the last inequality we have used (2.8). Putting together these results, we conclude that: Corollary 2.6. To a fiber non-degenerate almost Dirac structure L on a fibration p : M → B there is associated the following data:

• A connection ΓL on p : M → B. 2 • A horizontal 2-form ωL ∈ Ω (Hor ). 2 • A vertical bivector field πL ∈ X (Vert ). Conversely, every such triple (Γ, ω, π) on a fibration p : M → B determines a unique fiber non-degenerate almost Dirac structure L, which is given by:

(2.9) L = graph (πL) ⊕ graph (ωL). 2.3. Fiber non-degenerate Dirac structures. The next natural question is: Given a fiber non-degenerate almost Dirac structure L on a fibration p : M → B, what are the conditions on the associated triple (ΓL, ωL, πL) that guarantee that L is integrable, i.e., is a Dirac structure? Recall (see [5]) that the obstruction to integrability for an almost Dirac struc- 3 ture L is a 3-form TL ∈ Ω (L), which is defined on sections s1, s2, s3 ∈ Γ(L) by:

(2.10) TL(s1, s2, s3) := h s1, s2 , s3i+ J K where: • ·, · denotes the Courant bracket, on X(M) ⊕ Ω1(M), given by: J K (2.11) (X, α), (Y, β) := ([X,Y ], LX β − LY α + dh(X, α), (Y, β)i−) J K 1 • h·, ·i+ denotes the natural pairing on X(M) × Ω (M), defined by: 1 (2.12) h(X, α), (Y, β)i := (i α + i β) . + 2 Y X Our next result gives the 3-form TL of a fiber non-degenerate almost Dirac structure L in terms of the geometric data (ΓL, πL, ωL). We need to introduce some notation. For a vector field v ∈ X(B) we denote by ve ∈ X(M) its horizontal lift, and we # ∗ let πL : Vert → Vert denote the bundle map induced by the vertical bivector field πL. Also, we have isomorphisms: Hor ' L ∩ (TM ⊕ Vert 0), Hor 0 ' L ∩ (Vert ⊕ T ∗M). These allow us to identify a horizontal vector field X ∈ Γ(Hor ) with a section 0 0 sX = (X, α) ∈ Γ(L), where α ∈ Γ(Vert ), and a vertical form β ∈ Γ(Hor ) with a section sβ = (Y, β) ∈ Γ(L), where Y ∈ Γ(Vert ). In this notation we have: POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 7

Proposition 2.7. Let (ΓL, πL, ωL) be the geometric data determined by a ﬁber non-degenerate almost Dirac structure L on a ﬁber bundle p : M → B. Then: (i) If α, β, γ ∈ Γ(Hor 0) then: 1 T (s , s , s ) = [π , π ](α, β, γ). L α β γ 2 L L (ii) If v ∈ X(B) and β, γ ∈ Γ(Hor 0), then: 1 T (se, s , s ) = Leπ (β, γ). L v β γ 2 v L 0 (iii) If v1, v2 ∈ X(B) and γ ∈ Γ(Hor ), then: 1 T (se , se , s ) = (γ(Ω (v , v )) + π (die ie ω , γ)) L v1 v2 γ 2 ΓL e1 e2 L v1 v2 L

where ΩΓL is the curvature 2-form of ΓL. (iv) If v1, v2, v3 ∈ X(B), then: 1 T (se , se , s e ) = d ω (v , v , v ), L v1 v2 v3 2 ΓL L e1 e2 e3 • •+1 where dΓL :Ω (Hor ) → Ω (Hor ) is the diﬀerential induced by ΓL. The proofs are routine calculations so we omit them. Now observe that sections of the form sv˜ and sα with v ∈ X(B) and α ∈ Γ(Hor 0) generate Γ(L), as a C∞(M)-module. Therefore, as a corollary, we obtain the conditions that the triple (ΓL, πL, ωL) must satisfy for the associated L to be a Dirac structure:

Corollary 2.8. Let (ΓL, πL, ωL) be the geometric data determined by a fiber non-degenerate almost Dirac structure L on a fiber bundle p : M → B. Then L is Dirac iff the following conditions hold:

(i) πL is a vertical Poisson structure: [πL, πL] = 0. (ii) Parallel transport along ΓL preserves the vertical Poisson structure πL:

LveπL = 0, ∀v ∈ X(B).

(iii) The horizontal 2-form ωL is closed: dΓωL = 0. (iv) The following curvature identity is satisfied: # (2.13) ΩΓ(v1, v2) = πL (dive1 ive2 ωL), ∀v1, v2 ∈ X(B). The curvature identity (2.13) expresses the fact that the curvature 2-form of the connection associated with a fiber non-degenerate Dirac structure L takes values in the vertical Hamiltonian vector fields. We will explore this property later in our study of Poisson fibrations.

2.4. Presymplectic forms and Poisson structures. Let us illustrate the previous results with the two extreme cases of Dirac structures determined by Poisson and presymplectic structures.

If L is determined by a presymplectic form, one checks easily: Proposition 2.9. Let Ω be a presymplectic form on the total space of a fibration p : M → B. Then L = graph (Ω) is a fiber non-degenerate Dirac structure iff the pull-back of Ω to each fiber is non-degenerate. 8 OLIVIER BRAHIC AND RUI LOJA FERNANDES

In this case, the vertical Poisson structure πL is non-degenerate on the fibers and coincides with the inverse of the restriction of Ω to the fibers. The converse is also true: a fiber non-degenerate Dirac structure L for which the vertical Poisson structure πL is non-degenerate on the fibers, is determined by a presymplectic form Ω. In fact, it follows from (2.9) that: −1 Ω = ωL ⊕ (πL) . Hence, fiber non-degenerate presymplectic forms are presymplectic forms which restrict to symplectic forms on the fibers.

Dually, for Poisson structures, it is also immediate to check: Proposition 2.10. Let Π denote a Poisson structure on the total space of a fibration p : M → B. Then L = graph (Π) is a fiber non-degenerate Dirac structure 0 0 iff Π is horizontal non-degenerate, i.e., Π|Vert 0 : Vert × Vert → R is a non- degenerate bilinear form.

In this case, the horizontal 2-form ωL is non-degenerate: in fact, Π gives an 0 isomorphism Vert → Hor , and under this isomorphism ωL coincides with the restriction Π|Vert 0 . The converse is also true: a ﬁber non-degenerate Dirac structure L for which the horizontal 2-form ωL is non-degenerate, is a Poisson structure Π. In fact, it follows from (2.9) that: −1 Π = (ωL) ⊕ πL. Hence, ﬁber non-degenerate Poisson structures are the same thing as the horizontal non-degenerate Poisson structures of Vorobjev ([15]).

3. Poisson fibrations In this section we will study Poisson fibrations and their relationship to fiber non-degenerate Dirac structures.

3.1. Poisson and symplectic fibrations. Let (F, π) be a Poisson manifold. We denote by Diff π(F ) the group of Poisson diffeomorphisms of F . This is the subgroup of Diff (F ) formed by all diffeomorphisms φ : F → F such that:

(φ)∗π = π. We are interested in the following class of fibrations: Definition 3.1. A Poisson fibration p : M → B is a locally trivial fiber bundle, with fiber type a Poisson manifold (F, π) and with structure group a subgroup G ⊂ Diff π(F ). More precisely, p is a submersion such that there exists a −1 covering {Ui}i of B, and trivializations φi : p (Ui) → Ui × F , with transition −1 functions φj ◦ φi |{x}×F , x ∈ Ui ∩ Uj belonging to G. When π is symplectic the fibration is called a symplectic fibration. If p : M → B is a Poisson fibration modeled on a Poisson manifold (F, π), each −1 fiber Fb carries a natural Poisson structure πb: if φi : p (Ui) → Ui × F is a local trivialization, πb is defined by: −1 πb = (φi(b) )∗π, POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 9 for b ∈ Ui. It follows from the definition that this 2-vector field is independent of the choice of trivialization. Note that the Poisson structures πb on the fibers can be glued to a Poisson structure πV on the total space of the fibration:

πV (x) = πp(x)(x), (x ∈ M). 2 2 This 2-vector field is vertical: πV takes values in ∧ Vert ⊂ ∧ TM. In this way, the fibers (Fb, πb) become Poisson submanifolds of (M, πV ). Example 3.2. An important class of Poisson fibrations is obtained as follows. Take any Poisson manifold (P, Π), fix a closed symplectic leaf B of P , and let p : M → B be a tubular neighborhood of B in P . Each fiber carries a natural Poisson structure, namely, the transverse Poisson structure ([17]). These transverse Poisson structures are all Poisson diffeomorphic and it follows from the Weinstein splitting theorem that this is a Poisson fibration. More generally, one can take any Dirac manifold (P,L) and fix a closed presymplectic leaf B of P . If p : M → B is a tubular neighborhood of B in P , then each fiber carries a natural Poisson structure, called also the transverse Poisson structure ([8]). These transverse Poisson structures are all Poisson diffeomorphic and it follows from the generalization of the Weinstein splitting theorem in [8] that this is a Poisson fibration. We saw in the previous section than any fiber non-degenerate Dirac structure L on the total space of a fibration p : M → B induces Poisson structures on the fibers. In fact, we have the following: Proposition 3.3. Let p : M → B be a fibration with connected base and compact fibers. If L is a fiber non-degenerate Dirac structure, then p : M → B admits the structure of a Poisson fibration such that πV = πL. Proof. Corollary 2.8 gives the vertical Poisson structure. While trivializations are obtained by parallel transport along paths in B, induced by the connection (F being any chosen fiber). Compactness of the fibers here just ensures completeness of horizontal lifts. Note that a Dirac structure on the total space of a fibration p : M → B for which the fibers are Poisson-Dirac submanifolds may fail to be a Poisson fibration. In other words, Proposition 3.3 becomes false if one omits the assumption of fiber non-degeneracy. This is illustrated by the following simple example.

Example 3.4. Take the fibration p : R3 → R obtained by projection on the x-axis and the Poisson bracket on R3 defined by: {x, y} = {x, z} = 0, {y, z} = x. Then each fiber is a Poisson-Dirac submanifold: the fiber over x = 0 has the zero Poisson structure, while the fibers over x 6= 0 are symplectic. Since the fibers are not Poisson diffeomorphic, p : R3 → R cannot be a Poisson fibration. 3.2. Coupling Dirac structures. Motivated by Proposition 3.3 we introduce the following definition: Definition 3.5. If p : M → B is a Poisson fibration, we will say that a fiber non-degenerate Dirac structure L is compatible with the fibration if πL = πV . In this case, we call L a coupling Dirac structure. 10 OLIVIER BRAHIC AND RUI LOJA FERNANDES

Note that in the special case of a symplectic fibration, by the results of Section 2.4, a coupling Dirac structure is necessarily given by a presymplectic form Ω. In this case, Proposition 3.3 is well-known (see [13], Lemma 6.2). Moreover, given a symplectic fibration p : M → B, there are well-known non-trivial obstructions for the existence of a coupling 2-form ω ∈ Ω2(M). Our purpose now is to determine the corresponding obstructions for a general Poisson fibration and answer the following question: • Given a Poisson fibration p : M → B, is there a coupling Dirac structure L compatible with the fibration? Let us recall the notion of a Poisson connection:

Definition 3.6. A connection Γ on a Poisson ﬁbration p : M → B is called a Poisson connection if, for every path γ, parallel transport

φγ :(Fγ(0), πγ(0)) → (Fγ(1), πγ(1)) is a Poisson diffeomorphism. Clearly, a connection Γ on a Poisson fibration p : M → B is Poisson iff

LveπV = 0, ∀v ∈ X(B). Hence, by Corollary 2.8 (ii), an obvious necessary condition for the existence of a coupling Dirac structure is the existence of a Poisson connection. However, one can show that a Poisson ﬁbration always admits such a connection. In fact, we have:

Proposition 3.7. Let p : M → B be a Poisson ﬁbration. There exists a ﬁber non-degenerate almost Dirac structure L on p : M → B such that:

(a) The vertical bivector field πL coincides with πV . (b) The connection ΓL is a Poisson connection. Proof. Let p : M → B be a Poisson fibration with fiber (F, π) and choose −1 choose local trivializations φi : p (Ui) → Ui × F . Let Li be the Dirac structure on Ui × F obtained by pull-back of Lπ = graph (π) under the projection Ui × F → F : ∗ Li := {((v, w), (0, η)) ∈ T (Ui × F ) ⊕ T (Ui × F ): w = π(η, ·)}.

Observe that to Li it is associated the geometric data (ΓLi , πLi , ωLi ) where ΓLi is the canonical ﬂat connection on Ui × F → Ui, πLi = π and ωLi = 0. Hence Li is

fiber non-degenerate, induces π on the fibers, and ΓLi is a Poisson connection. Next, we choose a partition of unity ρi : B → R subordinated to the cover {Ui}, and we define X ∗ L := (ρi ◦ p)φi Li. i

By this we mean that the associated geometric data (ΓL, πL, ωL) has connection P ∗ P ∗ ΓL = i(ρi ◦ p)φi ΓLi , vertical bivector ﬁeld πL = i(ρi ◦ p)φi πLi and horizontal P ∗ 2-form ωL = i(ρi ◦ p)φi ωLi . It is clear that L is a ﬁber non-degenerate almost Dirac structure. Moreover, πL coincides with πV . Finally, given v ∈ X(B), if vei denotes the horizontal lift ∗ relative to φi ΓLi , we have:

Lvei πV = 0. POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 11

P The horizontal lift of v relative to ΓL is ve = i(ρi ◦ p)vei. It follows that: LveπV = [v,e πV ] X = [(ρi ◦ p)vei, πV ] i X # = (ρi ◦ p)[vei, πV ] + πV (d(ρi ◦ p)) i X = (ρi ◦ p)Lvei πV = 0, i 0 where we have used the fact that d(ρi ◦ p) ∈ Vert . Hence L is the desired almost Dirac structure. There is also a procedure to construct coupling Dirac structures due to A. Wade ([16]), which is entirely analogous to a construction in symplectic geometry due to A. Weinstein (see [13, Theorem 6.17]): Theorem 3.8. Let G × F → F be a Hamiltonian action of a compact Lie group G on the Poisson manifold (F, π). Every connection on a principal G-bundle P → B determines a coupling Dirac structure L on the associated Poisson fibration P ×G F → B. Proof. The connection Γ on P determines a projection TP → Vert , along the horizontal distribution Hor . Hence, there is an injection: ∗ ∗ iΓ : Vert ,→ T P, ∗ ∗ where Vert is the vertical cotangent bundle with typical fiber T Pb. Since Γ is G- invariant, the inclusion is G-equivariant. Therefore, the canonical symplectic form ∗ ∗ ωcan in T P induces a closed 2-form on Vert ∗ ωΓ = iΓωcan, which is G-invariant and restricts to the canonical symplectic form on the fibers ∗ ∗ ∗ ∗ T Pb. Also, the action of G on Vert has a moment map µP ◦ iΓ : Vert → g , ∗ ∗ where µP is the moment map of the lifted cotangent action G × T P → T P . ∗ Now let G × F → F be a Hamiltonian action with moment map µF : F → g . This determines a Hamiltonian G-action on the Dirac manifold M := Vert ∗ × F , with Dirac structure L = graph (ωΓ) ⊕ graph (π). and with moment map µM = (µP ◦ iΓ) ⊕ µF .

This action is free and 0 is a regular value of µM . Therefore, the reduced space −1 Mred := µM (0)/G ' P ×G F, carries a Dirac structure. It is easy to check that this Dirac structure is fiber non- degenerate and restricts to the canonical Poisson structures on the fibers. Hence, L is the desired coupling Dirac structure. Remark 3.9. Note that the resulting coupling Dirac structure is presymplectic iff the fiber (F, π) is symplectic. Also, it is easy to check that the remaining geometric data associated with the coupling Dirac structure L in the theorem above is the following: 12 OLIVIER BRAHIC AND RUI LOJA FERNANDES

• The connection ΓL is just the connection on the ﬁber bundle P ×G F induced from the connection Γ on P . • The horizontal 2-form ωL is given by the curvature of the connection composed with the moment map

ωL(˜v1, v˜2)([u, x]) = hµF (x),FΓ(˜v1, v˜2)ui, ([u, x] ∈ P ×G F ). Hence, the resulting coupling Dirac structure is Poisson iff the curvature 2-form of the connection is non-degenerate on the image of µF . Such a connection is sometimes called a fat connection. The proof of A. Wade in [16] consists in proving that this data satifies the conditions of Corollary 2.8 and so defines a coupling. 3.3. Obstruction to the existence of coupling. The construction of Theo- rem 3.8 can be extended for general Poisson fibrations, leading to a characterization of those fibrations which admit a coupling Dirac structure. Let L be a fiber non-degenerate almost Dirac structure on a Poisson fibration p : M → B. We will say that L is compatible with the fibration if πL = πV . By Proposition 3.7, any Poisson fibration admits a compatible L such that ΓL is a Poisson connection. For L to be Dirac this connection must have a much more constrained holonomy: Theorem 3.10. Let p : M → B be a Poisson fibration and let L be a compatible almost Dirac structure such that ΓL is a Poisson connection. Then the following statements are equivalent:

(i) L is a Dirac structure: TL = 0. (ii) For every base point b ∈ B, the action of the holonomy group Φ(b) of ΓL on the fiber Fb is Hamiltonian. A proof of this result will be given in the next paragraph, using a Poisson gauge theory. An immediate corollary is the following: Corollary 3.11. Let (F, π) be a compact Poisson manifold whose first Poisson 1 cohomology group vanishes: Hπ(F ) = 0. Then any Poisson fibration p : M → B with fiber (F, π), and finite dimensional structure group admits a coupling Dirac structure.

1 Proof. Since Hπ(F ) = 0, the same holds for the fibers Fb, and it follows that any Poisson action on the fibers is Hamiltonian. Now apply Proposition 3.7 to construct a fiber non-degenerate almost Dirac structure L compatible with the fibration and such that ΓL is a Poisson connection. Using the implication (ii) ⇒ (i) in Theorem 3.10, we conclude that L is a coupling Dirac structure. Compacity of F ensures completeness of the connection.

Notice that the condition that the holonomy group Φ(b) of ΓL acts in a Hamil- tonian fashion on the fiber Fb is a property of its connected component of the identity Φ(b)0. This connected component is known as the restricted holonomy group and is formed by the holonomy homomorphisms φγ , where γ is a contractible loop based at b. In particular, each φγ with γ a contractible loop based at b, lies in the group of Hamiltonian diffeomorphisms Ham (Fb, πb), which is known to be a normal subgroup of the group of Poisson diffeomorphisms Diff (Fb, πb). The quotient group Diff (Fb, πb)/Ham (Fb, πb) is known as the group of outer Poisson diffeomorphisms. POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 13

We conclude that a coupling Dirac structure L for a Poisson ﬁbration p : M → B has an associated coupling holonomy homomorphism:

φ : π1(B, b) → Diﬀ (Fb, πb)/Ham (Fb, πb), [γ] 7→ [φγ ]. Example 3.12. A tubular neighborhood p : M → B of a symplectic leaf B of a Poisson manifold (P, Π) (see Example 3.2) admits LΠ as a coupling Dirac structure. It follows that the connection ΓΠ is Poisson and has Hamiltonian holonomy around any contractible loop in B. This can also be proved directly using the Weinstein splitting theorem. In general, the holonomy around a non-contractible loop will not be Hamilton- ian and we will have a nontrivial homomorphism

φ : π1(B) → Diff (F, π)/Ham (F, π). This is precisely the (reduced) Poisson holonomy of the leaf B introduced in [6]. 3.4. Poisson Gauge Theory. We now turn to the proof of Theorem 3.10. The idea will be to give an analogue of Theorem 3.8, but where the structure group is allowed to be infinite dimensional. We consider a Poisson fibration p : M → B with fiber type a Poisson manifold (F, π). The structure group of this fibration is the group G = Diff (F, π) of Poisson diffeomorphisms. The corresponding principal G-bundle is the Poisson frame bundle: P → B whose fiber over a point b ∈ B is formed by all Poisson diffeomorphisms u : F → Fb. The group G acts on (the right of) P by pre-composition: P × G → P :(u, g) 7→ u ◦ g. Then our original Poisson fiber bundle is canonically isomorphic to the the associated fiber bundle: M = P ×G F . Every Poisson connection Γ on the Poisson fiber bundle p : M → B is induced by a principal bundle connection on P → B. To see this, observe that the tangent ∞ ∗ space TuP ⊂ C (u TM) at a point u ∈ P is formed by the vector fields along u, X(x) ∈ Tu(x)M such that:

du(x)p · X(x) = constant, LX πV = 0. The Lie algebra g of G is the space of Poisson vector ﬁelds: g = X(F, π). The inﬁnitesimal action on P is given by:

ρ : g → X(P ), ρ(X)u = du · X, so the vertical space of P is:

Vert u = {du · X : X ∈ X(F, π)} . Now a Poisson connection Γ on p : M → B determines a connection in P → M whose horizontal space is:

Hor u = {ve ◦ u : v ∈ TbB} , where u : F → F and v : F → T M denotes the horizontal lift of v. Clearly, this b e b Fb defines a principal bundle connection on P → B, whose induced connection on the associated bundle M = P ×G F is the original Poisson connection Γ. Fix a Poisson connection Γ on the Poisson fiber bundle p : M → B. Recall that the holonomy group Φ(b) with base point b ∈ B is the group of holonomy 14 OLIVIER BRAHIC AND RUI LOJA FERNANDES transformations φγ : Fb → Fb, where γ is a loop based at b. Clearly, we have Φ(b) ⊂ Diff (Fb, πb). On the other hand, for u ∈ P we have the holonomy group Φ(u) ⊂ G = Diff (F, π) of the corresponding connection in P which induces Γ: it consist of all elements g ∈ G such that u and ug can be joined by a horizontal curve in P . Obviously, these two groups are isomorphic, for if u : F → Fb then: Φ(u) → Φ(b), g 7→ u ◦ g ◦ u−1, is an isomorphism. The curvature of a principal bundle connection is a g-valued 2-form FΓ on P which transforms as: ∗ −1 RgFΓ = Ad (g ) · FΓ, (g ∈ G).

Therefore, we can also think of the curvature as a 2-form ΩL with values in the adjoint bundle gP := P ×G g. In the case of the Poisson frame bundle, the adjoint bundle has fiber over b the space X(Fb, πb) of Poisson vector fields on the fiber. Hence the curvature of our Poisson connection can be seen as a 2-form ΩΓ : TbB × TbB → X(Fb, πb). The two curvature connections are related by: −1 (3.1) ΩΓ = du ◦ FΓ ◦ u . Finally, it is easy to check that, in fact, we have:

ΩΓ(v1, v2) = [ve1, ve2] − [^v1, v2], which is the expression we have used before for the curvature. After these preliminarities, we can now proceed to the proof. Proof of Theorem 3.10. We will prove the two implications separately. (i) ⇒ (ii). Let us start by observing that given any u ∈ P , a Poisson diffeomorphism u : F → Fb, the curvature identity (2.13) together with (3.1) shows that, for any v1, v2 ∈ TbB, the vector field FΓ(v1, v2)u ∈ g = X(F, π) is Hamiltonian: # FΓ(v1, v2)u = π d(ωL(ve1, ve2) ◦ u). Now fix u0 ∈ P . The Holonomy Theorem states that the Lie algebra of the holonomy group Φ(u0) is generated by all values FΓ(v1, v2)u, with u ∈ P any point that can be connected to u0 by a horizontal curve. Hence, we can define a moment ∗ map µF : F → (Lie(Φ(u0)) ) for the action of Φ(u0) on F by:

(3.2) hµF (x),FΓ(v1, v2)ui = ωL(ve1, ve2)u(x). This shows that the action of Φ(u0) on (F, π) is Hamiltonian, and so (ii) holds (recall the comments above about the relationship between the holonomy groups Φ(b) and Φ(u)).

(ii) ⇒ (i). Again we fix u0 ∈ P , and we assume now that the action of Φ(u0) on ∗ (F, π) is Hamiltonian with moment map µF : F → (Lie(Φ(u0)) ). By the Reduction Theorem we can reduce the principal Poisson frame bundle to a principal Φ(u0)- bundle P 0 → B. Now we can apply (the infinite dimensional version) of Theorem 3.8 to produce a coupling Dirac structure on the associated Poisson fiber bundle p : M → B. Instead, if the reader does not like an infinite dimensional argument, he can check by himself that the geometric data formed by the connection ΓL, the vertical Poisson vector field πV and the 2-form ωL defined from (3.2) (we are now given µF and we define ωL) satisfy the conditions of Corollary 2.8. POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 15

Remark 3.13. Note that our proof really shows that for any Poisson ﬁbration the coupling Dirac structure arises as in the construction of Theorem 3.8. Rela- tion (3.2) between the moment map µF , the curvature of the connection, and the horizontal 2-form ωL, was already present there (see Remark 3.9).

4. Integration of Poisson fibrations In this section, we study the integration of Poisson fibrations. Just as Poisson manifolds integrate to symplectic groupoids, we will see that Poisson fibrations integrate to fibered symplectic groupoids.

4.1. Fibered symplectic groupoids. Recall that for us a fibration always means a locally trivial fiber bundle. If we fix a base B, we have a category Fib of fibrations over B, where the objects are the fibrations p : M → B and the morphisms are the fiber preserving maps over the identity:

φ M1 / M2 B BB || BB || p1 BB || p2 B ~|| B A fibered groupoid is an internal groupoid in Fib, i.e., an internal category where every morphisms is an isomorphism. This means that both the total space G and the base M of a fibered groupoid are fibrations over B and all structure maps are fibered maps. For example, the source and target maps are fiber preserving maps over the identity: G / M ? / ?? }} ? }} ?? }} ? ~}} B In particular, each fiber of G → M is a groupoid over a fiber of M → B. Moreover, the orbits of G lie inside the fibers of the base M. A general procedure to construct fibered Lie groupoids is a follows. Let P → B be a principle G-bundle and assume that G acts on a groupoid F ⇒ F by groupoid automorphisms. Then the associated fiber bundles G = P ×G F and M = P ×G F are the spaces of arrows and objects of a fibered Lie groupoid. Clearly, every fibered Lie groupoid is of this form provided we allow infinite dimensional structure groups. We will say that the fibered Lie groupoid G has fiber type the Lie groupoid F. Definition 4.1. A fibered symplectic groupoid is a fibered Lie groupoid G whose fiber type is a symplectic groupoid (F, ω). Therefore, if G is a fibered symplectic groupoid over B, then G → B is a symplectic fibration, and each symplectic fiber Fb is in fact a symplectic groupoid over the corresponding fiber Fb of M → B.

Proposition 4.2. The base M → B of a fibered symplectic groupoid G ⇒ M has a natural structure of a Poisson fibration. Proof. Note that (i) the base of any symplectic groupoid has a natural Poisson structure for which the source (respectively, the target) is a Poisson (respectively, 16 OLIVIER BRAHIC AND RUI LOJA FERNANDES anti-Poisson) map, and (ii) any symplectic groupoid isomorphism between two symplectic groupoids covers a Poisson diffeomorphism of the base Poisson manifolds. Hence, each fiber of the base M → B of a fibered symplectic groupoid carries a natural Poisson structure, and a trivialization of the fibered symplectic groupoid covers a trivialization of M → B whose transition functions are Poisson diffeomorphisms of the fibers. Therefore the result follows. Note that the fibers p : M → B are integrable Poisson manifolds. 4.2. Integration of Poisson fibrations. We just saw that the base of a fibered symplectic groupoid is a Poisson fibration. We will say that a fibered symplectic groupoid G → B integrates a Poisson fibration p : M → B whenever this fibration is (Poisson) isomorphic to the Poisson fibration determined by G → B. If such a a fibered symplectic groupoid exists we say that the Poisson fibration is integrable. Note that the fiber type F of G is a a symplectic groupoid integrating the fiber type (F, π) of p : M → B. Theorem 4.3. A Poisson fibration is integrable iff its fiber type is an integrable Poisson manifold. There exists a 1:1 correspondence between, source 1-connected, fibered symplectic groupoids and integrable Poisson fibrations. In one direction, the proof follows from Proposition 4.2. For the other direction, we will offer two proofs. The first proof is an heuristic proof that uses Poisson gauge theory. The second proof uses the approach to integrability through cotangent paths developed in [3, 4]. Heuristic proof via gauge theory. Given a Poisson fibration p : M → B with fiber type (F, π), we start by writing it as an associated fiber bundle:

M = P ×G F where P is the Poisson frame bundle and G ⊂ Diff (F, π) is the structure group of the fibration. Since (F, π) is integrable, there exists a unique source 1-connected symplectic groupoid F ⇒ F which integrates (F, π). The action of G on F lifts to an action of G on F by symplectic groupoid automorphisms (see [7]). Hence, we can form the associated bundle: G = P ×G F. Since the action of G on F is by groupoid automorphisms, G → B becomes a fibered groupoid over M → B. Since this action is by symplectomorphisms, G → B becomes a symplectic fibration. Since the groupoid structures and the symplectic structure on the fibers are compatible, G is a source 1-connected fibered symplectic groupoid with fiber type F. It should be clear that the Poisson fibration determined by G is isomorphic to the original fibration. Remark 4.4. Note that this heuristic proof becomes a real proof if the structure group G of the fibration is a finite dimensional Lie group. We will illustrate this below in Example 4.3. Proof of Theorem 4.3. Given a Poisson fibration p : M → B with fiber type (F, π), we denote by Σ(M) ⇒ M the symplectic groupoid that integrates the vertical Poisson structure πV . Note that since we assume that (F, π) is integrable, we have that (M, πV ) is integrable, so that Σ(M) is a Lie groupoid. POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 17

Let us recall (see [3, 4] for details and notations) that Σ(M) is the space of equivalence classes of cotangent paths: {a : [0, 1] → T ∗M : π] (a(t)) = d p(a(t))} Σ(M) = V dt ; {cotangent homotopies} where p : T ∗M → M is the cotangent bundle projection. Now we observe that Vert 0 ⊂ T ∗M is a Lie subalgebroid, which is in fact, a bundle of Abelian Lie algebras. This is a direct consequence of πV being vertical. Hence, the equivalence classes of cotangent paths with image in Vert 0 form a closed Lie subgroupoid K ⊂ Σ(M), which is in fact a bundle of Abelian Lie groups. Let us consider the quotient Lie groupoid: G := Σ(M)/K. Notice that G is fibered over B, where the fiber over b is the symplectic groupoid Σ(Fb) integrating the fiber (Fb, πb). It is easy to check that G is, in fact, the desired source 1-connected, fibered symplectic groupoid integrating p : M → B. We leave the details to the reader. As we have seen in Proposition 3.7, a Poisson fibration p : M → B always admits Poisson connections. What does the specification of a Poisson connection on p : M → B amounts to in the corresponding fibered symplectic groupoid G? Proposition 4.5. Let M → B be a Poisson fibration which integrates to a source 1-connected fibered symplectic groupoid G → B. The choice of a Poisson connection on the Poisson fibration M → B determines a coupling form Ω on the fibered symplectic groupoid G → B, and conversely. Proof. We will give a “gauge theoretical” proof, which is valid at least in the case where the structure group is a finite dimensional Lie group. One can also give a longer proof using paths, which avoids this assumption. Hence, assume that:

M = P ×G F, where P is a principal G-bundle and F is a Poisson G-space. As we have men- tioned above, the Poisson action G × F → F lifts to an action G × F → F by automorphisms of the symplectic groupoid F = Σ(F ), which is Hamiltonian with equivariant moment map J : F → g∗, which is a groupoid cocycle. We have that:

G = P ×G F. Now, Poisson connections Γ on M → B are in 1:1 correspondence with principal bundle connections on P . To complete the proof we observe that, since the action G × F → F is Hamil- tonian, a choice of a principal bundle connection on P determines a coupling form on G and conversely (see Theorem 3.8 or [13, Chapter 6] for more details). The next natural question is: what does a coupling Dirac structure on the Poisson ﬁbration p : M → B amounts to in the corresponding ﬁbered symplectic groupoid G? This question is more delicate, and it is intimately related with the pre-symplectic groupoids integrating Dirac structures described in [2]. This will be discussed elsewhere. 18 OLIVIER BRAHIC AND RUI LOJA FERNANDES

4.3. An Example. Let us denotes by S3 → S2 the Hopf ﬁbration which we view as a principal S1-bundle P → S2. We will consider as ﬁber types (F, π) the following two Poisson S1-manifolds: 1) The manifold F = S2, with the standard area form and the S1-action by rotations around the north-south poles axis; 2) The manifold F = su(2)∗ ' R3, with its canonical linear Poisson structure and the S1-action by rotations around the z-axis;

1 The corresponding Poisson fibrations M = P ×S F are: 1) the non-trivial S2-bundle p : M → S2 (a symplectic fibration), and 2) the non-trivial rank 3 vector bundle p : E → S2 (a Poisson fibration which is not symplectic). The symplectic leaves of su(2)∗ are the concentric spheres around the origin and the origin itself. Hence the Poisson fibration p : E → S2 is foliated by symplectic fibrations isomorphic to p : M → S2 and the zero section. Since S2 is symplectic, we have: 1 2 1 2 Hπ(S ) ' H (S ) = {0}. Since su(2) is semisimple of compact type, we also have:

1 ∗ Hπ(su(2) ) = {0}.

It follows from Corollary 3.11 that both p : E → S2 and p : M → S2 admit coupling Dirac structures. Of course, since p : M → S2 is a symplectic fibration, its Dirac coupling is actually associated with a closed 2-form. We let the reader check that the presymplectic leaves of the Dirac coupling for p : E → S2 are the symplectic fibrations isomorphic to p : M → S2 (with their coupling forms) and the zero section (with the zero 2-form). Let us now turn to the fibered symplectic groupoids integrating these fibrations. For that, we use the method in the heuristic proof of Theorem 4.3. Since the structure group is S1, a finite dimensional Lie group, this is allowed. We need the source 1-connected symplectic groupoid F = Σ(F ) integrating the fiber type, and this is well-known in both examples: 1) Since S2 is symplectic and 1-connected, the associated source 1-connected symplectic groupoid is the pair groupoid Σ(S2) = S2 × S2, where the bar over the second factor means that we change the sign of symplectic form. 2) From general facts about linear Poisson structures, the symplectic groupoid of su(2)∗ is Σ(su(2)∗) = T ∗SU(2), furnished with the canonical cotangent bundle symplectic structure. This groupoid is isomorphic to the action groupoid SU(2) n su(2)∗ for the coadjoint action of SU(2) on su(2)∗. Now we can describe, in both cases, the fibered symplectic groupoid G → S2 given by Theorem 4.3. For the non-trivial S2-fibration M → S2, the action of S1 on S2 lifts to the diagonal S1-action on S2 × S2, and we have: 2 1 2 G(M) = P ×S (S × S ), which is a non-trivial symplectic (S2 × S2)-fibration over S2 and a groupoid over the non-trivial S2-fibration. POISSON FIBRATIONS AND FIBERED SYMPLECTIC GROUPOIDS 19

For the rank 3 vector bundle E → S2, the action of S1 on su(2)∗ lifts to an action on SU(2) × su(2)∗, which is trivial on the first factor, and we have: ∗ 1 G(E) = P ×S (SU(2) × su(2) ) ' E × SU(2). Note that, contrary to the case of the Poisson fibrations, the symplectic fibered groupoid G(M) does not sit naturally in G(E). This is because a Poisson submanifold does not always integrate to a symplectic subgroupoid, and this is exactly the case with the spheres in su(2)∗.

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Depart. de Matematica,´ Instituto Superior Tecnico,´ 1049-001 Lisboa, PORTUGAL E-mail address: [email protected], [email protected]